Adam Lore wrote:Keiji, your system is a great start. My only complaint is that it sounds very foreign.

Well, of course it does. It is based on Japanese, after all.

What if we somehow anglicized it (even more)?

for example, "cote" (sounds like "coat") instead of koto. Or "Cotone"..

Well the entire point of what I did was to make them nice and regular. If you changed koto into cote, teto (truncated tetrahedron) would become tet, which is the bowers acronym for the parent tetrahedron, which would get confusing. Cotone, on the other hand, sounds too much like cone.

Hmm, apparently I forgot to announce this here, but I've been making renders of the grand antiprism. Here's a sample showing the two rings of pentagonal antiprisms rotating in the plane of one of the rings:

I don't even know if anyone even reads this thread anymore, but since there's nowhere else to post it, here goes nothing:

This, my friends, is the perspective projection of the runcinated 120-cell into 3D showing the relationship between the 120 dodecahedra and the 600 tetrahedra. Cells on the far side of the polychoron have been culled for clarity's sake, as have the triangular and pentagonal prisms (of which there are 1200 and 720, respectively).

Of course, just this picture alone doesn't do justice to this beautiful uniform polychoron, so I account for every cell in the page dedicated to the runcinated 120-cell, showing them in 9 layers plus the equator. I have to say that the projections featured on that page are some of the most beautiful I've rendered yet. There are just triangular prisms and pentagonal prisms everywhere in all sorts of pentagonal, dodecahedral, and icosahedral patterns, seasoned with a sprinkling of tetrahedra and served with the right amount of dodecahedra. I enjoyed making them as much as I hope you'll enjoy looking at them. :-)

This is a perspective projection centered on one of the tetrahedra in the "second group" of tetrahedra. (The tetrahedra bounding the snub 24-cell can be divided into two groups, the first group of 96 which corresponds with the edges of the 24-cell from which it is derived, and the second group of 24 which corresponds with the vertices of said 24-cell. The second group tetrahedra only share a face with tetrahedra from the first group, whereas the tetrahedra from the first group also shares faces with the icosahedra.) Six icosahedra which share an edge with this tetrahedron are colored in 3 pairs, having octahedral symmetry. The remaining tetrahedra are rendered in (very) transparent yellow.

Hope y'all like it!

It took me a while to get this image just right... It's not simple to balance the color/transparency assignments so that what you want to emphasize is clearly visible. But after all, nobody said rendering 4D in 2D was easy.

Of course, being a perspective projection, this image only shows the "northern hemisphere" cells; the equatorial and southern cells have been omitted (since otherwise there would be such visual clutter that the image would be useless), and the first group tetrahedra aren't easily discerned (since emphasizing them will only obscure the other parts of the image); so for the full details, check out the snub 24-cell page. </shameless plug>

Well, I wasn't quite done for the day. :-P I was browsing over some of the older pages, and decided to give a few of them a face-lift, and among the updates is this brand new projection of the omnitruncated tesseract:

This one is centered on one of the truncated octahedra (instead of the usual great rhombicuboctahedron), which gives the projection a nice tetrahedral symmetry. You can see those big fat great rhombicuboctahedra squashed by the projection ('cos they're being viewed from an angle), almost like gemstones budding from the truncated octahedron. This one also took some effort to get right, but, I must say, I'm quite pleased with the result.

I seem to be spending more time in Tetronia this month than I've anticipated, but while I'm still here, I might as well send you poor trionians this nice family portrait of the convex regular polychora:

The 4 simpler polychora are projected vertex-first, while the 120-cell and the 600-cell are projected cell-first. (Just 'cos I can. ) You can look at the high resolution image for a clearer view of the highlighted vertices/cells.

This is more of a montage than a true family portrait, though. Each polychoron was individually photographed (er... projected) and then the photographs (er... projections) were assembled into a nice 3D scene. The nice thing about 4D stock paper is that I have a 3D amount of space to work with, so I can arrange for each individual portrait not to overlap with the rest while still taking up only minimal space. Of course, from you trionians' POV, they are overlapping, which is just a part of the illusion.

*Goes back to hide in Tetronia while Wendy laughs at me from 24D space. *

If this were intended for tetronians to look at, it'd be a 4D scene with the 6 polychora arranged in octahedral formation (since we have a 3D surface on our desk to put them on). Unfortunately, such an arrangement will probably be quite confusing for poor trionians, since the projection images of the polychora will overlap and partially obscure each other.

Having said that, though, I'm actually quite tempted to actually do such a render... once I get my polytope viewer to work with non-convex shapes, that is. Although, now that I think of it, there's a way to "hack" it by computing the orientation of the projection envelope and use that to do CSG in POVRay. Hmmm... maybe it is within easy reach after all! I shall have to think about this. Maybe it will work! (No guarantees that the result will be comprehensible to trionians, though! )

This is the truncated 600-cell. This is one of the prettiest renders I've made so far. It really shows you the symmetry of the upper half of the 120-cell (the icosahedra in the truncated 600-cell correspond with the dodecahedra of the 120-cell). You can see that surrounding the central icosahedron (magenta) are 12 icosahedra in dodecahedral symmetry (blue), then 32 more icosahedra (yellow) with 12 of them in dodecahedral symmetry and the other 20 in icosahedral symmetry. Not shown here are the equatorial icosahedra which correspond with the edges of the dodecahedron - but you can see them in a parallel projection on the truncated 600-cell page.

I wonder if a reverse perspective projection can be made on these spherical polychrons, cause one can gain a grasp of a 3-hemisphere with all 3 curvatures displayed as normal in such projections. IMO the curved 3D sheet that forms the 3-hemisphere will be more visible in such projection

Secret wrote:I wonder if a reverse perspective projection can be made on these spherical polychrons, cause one can gain a grasp of a 3-hemisphere with all 3 curvatures displayed as normal in such projections. IMO the curved 3D sheet that forms the 3-hemisphere will be more visible in such projection

In your perspective projection renders, the centermost/innermost cell is the cell closest to us in 4DWould you render a similar diagram but instead treating the centermost cell as the cell farthest from us in 4D and place it 'outside'.

In other words, can you treak the lighting so that the center cell in the projection is least obsurbed while the outermost layer (the spherequatorial cells) the most obsurbed?(might put pics to illustrate the idea in case you don't understand)

Cause (not sure if it is just wishful thinking/illusion/whatever) I think I can see 4d better that way

Funny you should mention that; I was thinking for a while now of making the nearest cell opaque, and subsequent layers more and more transparent. I think it helps me visualize 4D too. I'll try to write up a render script for, say, the 120-cell, and see if it turns out well.